MIG wire or solder that LOOKS like brazing?

Yes, you can use a CuAl-wire or a CuSi-wire (Al-bronze or Si-bronze). But they cost! The color is not as gold as brazing, and it looks more like a weld. Not the flowing of real brazing.
And then, you could use paint. :-)
Nick

MAPP gas, when used with a torch intended for MAPP, will allow you to braze or silver solder nails together. Brazing with a torch will create the flowed look, and the joints will be goldish/yellow brass-colored. A problem you might encounter, that you could probably overcome with practice, is that the MAPP torch will be likely heat a relatively large area, and you might have nearby joints coming apart as the heat spreads to previously made joints.
A problem with brazing small parts is keeping the pieces in place when your hands are full, one with the torch and the rod in the other. You may need to find or build something similar to a universally adjustable helping hand.
Soft solders would also work, if the finished pieces aren't handled too roughly. Soft solders will flow well and leave a silvery-grey color. Some people get freaky about lead used in common solders, so there might be a senseless issue with that.
I haven't used the brazing wire that's available for MIG welder use, so I can't comment about flow or appearance, although I trust Nick's comment about the cost of the wire is correct. When I looked for it online last year, I don't think I was able to find it in small spools.
WB ......... metalworking projects

An addition: Should you want to try out how it looks, chances are *very* high that a halfway decent body shop has the bronze-wires. Some (many? all? at least not none) car-manufacturers require repairs to be made with bronze. They have a smaller HAZ and the zinc that evaporates near the weld is less.
A beer or two might help to convince them to make a little sample weld for you.
Nick

I used some 56% silver braze last Saturday to piece a live tool guide ring together for an Index G200. I noticed that my silver braze took on a yellowish cast. Not as noticible as typical brazing rod but it did look nice. Normally I only fit edge to edge and never really see the silver braze but in this case there was a bit of the ring missing so I filled in one of the counter sinks where it broke. So far it is holding, replacement ordered and on the way.
Wes